Proof Complexity Generators

by Jan Krajícek

Overview

The P vs. NP problem is one of the fundamental problems of mathematics. It asks whether propositional tautologies can be recognized by a polynomial-time algorithm. The problem would be solved in the negative if one could show that there are propositional tautologies that are very hard to prove, no matter how powerful the proof system you use. This is the foundational problem (the NP vs. coNP problem) of proof complexity, an area linking mathematical logic and computational complexity theory. Written by a leading expert in the field, this book presents a theory for constructing such hard tautologies. It introduces the theory step by step, starting with the historic background and a motivational problem in bounded arithmetic, before taking the reader on a tour of various vistas of the field. Finally, it formulates several research problems to highlight new avenues of research.

Product Details

ISBN-13:	9781009611701
Publisher:	Cambridge University Press
Publication date:	06/26/2025
Series:	London Mathematical Society Lecture Note Series , #497
Pages:	134
Product dimensions:	5.98(w) x 8.98(h) x 0.31(d)

About the Author

Jan Krajíček is Professor of Mathematical Logic at Charles University, Prague. A member of the Learned Society of the Czech Republic and the Academia Europaea, he has previously published three books with Cambridge University Press (1995, 2011 and 2019).

Table of Contents

1. Introduction; 2. The dWPHP problem; 3. τ-formulas and generators; 4. The stretch; 5. Nisan-Wigderson generator; 6. Gadget generator; 7. The case of ER; 8. Consistency results; 9. Contexts; 10. Further research; Special symbols; References; Index.